Download GTU BE/B.Tech 2019 Winter 4th Sem New 2141905 Complex Variables And Numerical Methods Question Paper

Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2019 Winter 4th Sem New 2141905 Complex Variables And Numerical Methods Previous Question Paper

Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ? IV (New) EXAMINATION ? WINTER 2019
Subject Code: 2141905 Date: 07/12/2019

Subject Name: Complex Variables and Numerical Methods

Time: 10:30 AM TO 01:30 PM Total Marks: 70

Instructions:

1. Attempt all questions.

2. Make suitable assumptions wherever necessary.

3. Figures to the right indicate full marks.

Q-1 (a)
Determine whether the function {
?? 2
+3???? ?2
?? +?? ; ?? ? ??? 5 ; ?? = ??? is continuous?
Can the function be redefined to make it continuous at ?? = ??? ?
03
(b)
State De Moivre?s Theorem. Find the roots of the equation ?? 4
+ 1 = 0.
04
(c) Solve the following system of equations using Gauss Seidel Method correct to
four decimal places.
30?? ? 2?? + 3?? = 75; 2?? + 2?? + 18?? = 30; ?? + 17?? ? 2?? = 48
07

Q-2 (a)
Check whether the function ?? (?? ) = ?? ?? 2
is entire or not. Also find derivative of
?? (?? ).
03
(b)
Find the bilinear transformation which maps ?? = 1, 0, ?1 into the points ?? =
?? , ?, 1.
04
(c)
Using the Residue Theorem Evaluate, ?
????
5?3???????? 2?? 0

07
OR
(c) Show that the function ?? (?? , ?? ) = 3?? 2
?? + 2?? 2
? ?? 3
? 2?? 2
is harmonic. Find the
conjugate harmonic function ?? and express ?? + ???? as analytic function of z
07

Q-3 (a)
Evaluate ?
?? 2
+1
?? 2
?1
????
?? if ?? is the circle of unit radius with centre at ?? = 1.
03
(b)
Find the real part and imaginary part of ??? ???
04
(c) Evaluate ? ?? (?? )???? where ?? (?? ) is defined by
?? (?? ) = {
1 ?? ????? ?? < 0
4?? ?? ????? ?? > 0

And C is the arc from ?? = ?1 ? ?? to ?? = 1 + ?? along the curve ?? = ?? 3
.
07



OR

Q-3 (a)
Find the type of singularity of the function ?? (?? ) =
?? 2?? (?? ?1)
4

03
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Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ? IV (New) EXAMINATION ? WINTER 2019
Subject Code: 2141905 Date: 07/12/2019

Subject Name: Complex Variables and Numerical Methods

Time: 10:30 AM TO 01:30 PM Total Marks: 70

Instructions:

1. Attempt all questions.

2. Make suitable assumptions wherever necessary.

3. Figures to the right indicate full marks.

Q-1 (a)
Determine whether the function {
?? 2
+3???? ?2
?? +?? ; ?? ? ??? 5 ; ?? = ??? is continuous?
Can the function be redefined to make it continuous at ?? = ??? ?
03
(b)
State De Moivre?s Theorem. Find the roots of the equation ?? 4
+ 1 = 0.
04
(c) Solve the following system of equations using Gauss Seidel Method correct to
four decimal places.
30?? ? 2?? + 3?? = 75; 2?? + 2?? + 18?? = 30; ?? + 17?? ? 2?? = 48
07

Q-2 (a)
Check whether the function ?? (?? ) = ?? ?? 2
is entire or not. Also find derivative of
?? (?? ).
03
(b)
Find the bilinear transformation which maps ?? = 1, 0, ?1 into the points ?? =
?? , ?, 1.
04
(c)
Using the Residue Theorem Evaluate, ?
????
5?3???????? 2?? 0

07
OR
(c) Show that the function ?? (?? , ?? ) = 3?? 2
?? + 2?? 2
? ?? 3
? 2?? 2
is harmonic. Find the
conjugate harmonic function ?? and express ?? + ???? as analytic function of z
07

Q-3 (a)
Evaluate ?
?? 2
+1
?? 2
?1
????
?? if ?? is the circle of unit radius with centre at ?? = 1.
03
(b)
Find the real part and imaginary part of ??? ???
04
(c) Evaluate ? ?? (?? )???? where ?? (?? ) is defined by
?? (?? ) = {
1 ?? ????? ?? < 0
4?? ?? ????? ?? > 0

And C is the arc from ?? = ?1 ? ?? to ?? = 1 + ?? along the curve ?? = ?? 3
.
07



OR

Q-3 (a)
Find the type of singularity of the function ?? (?? ) =
?? 2?? (?? ?1)
4

03
(b)
Find and Sketch the region of an infinite strip 1 < ?? < 2 under the
transformation ?? =
1
??
04
(c)
Expand ?? (?? ) =
1
(?? ?1)(?? ?2)
valid for region
(i) |?? | < 1 (ii) 1 < |?? | < 2 (iii) |?? | > 2
07



Q-4 (a)
Use Euler?s Method, find ?? (0.2) given that
????
????
= ?? ? ?? 2
; ?? (0) = 1
take ? = 0.1
03

(b)
Evaluate ?8 to two decimal places by Newton?s iterative formula.
04

(c) Determine the polynomial by Newton?s forward difference formula from the
following table
?? 0 1 2 3 4 5
?? -10 -8 -8 -4 10 40

07

OR

Q-4 (a) Solve the following system ofequation using Gauss Elimination Method
?? + ?? + ?? = 7; 3?? + 3?? + 4?? = 24; 2?? + ?? + 3?? = 16
03
(b) Use Secant Method to find the root of ?? (?? ) = ?? ?????? 10
?? ? 1.9 = 0 04
(c) Using Newton?s Divided Differences formula to find a polynomial function,
satisfying the following data.
?? -4 -1 0 2 5
?? (?? ) 1245 33 5 9 1335

07



Q-5 (a)
Evaluate ?
????
1+?? 2
1
?1
by using Gaussian formula for ?? = 2 and ?? = 3
03
(b) Use fourth order Range-Kutta method to compute ?? (0.2) and ?? (0.4) given that
????
????
= ?? ?
2?? ?? ; ?? (0) = 1.
04
(c)
Find the dominant Eigen value of ?? = [
3 ?5
?2 4
] by Power Method and the
corresponding Eigen vector.
07
OR
Q-5 (a)
State Trapezoidal Rule and evaluate ? ?? ?? 1
0
???? using it with ?? = 10
03
(b) Use Lagrange?s formula to fit a polynomial to the data
?? -1 0 2 3
?? 8 3 1 12

04
(c) Apply improved Euler?s method to solve the initial value problem
?? ?
= ?? + ?? with ?? (0) = 0 choosing ? = 0.2 and compute ?? 1
, ?? 2
, ? , ?? 5
.
07
*********
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